Is there a function f(a,b) which maps ordered pairs to lines in a plane in a continuous, bijective manner?

Here is the definition I am using for the limit with lines: a sequence of lines $L(1), L(2), \dots$ is said to approach another line $L$ if, for any point $p$ on $L$, the limit as $n\to\infty$ of the distance between $p$ and $L(n)$ is 0.

If there is no such function, can anyone think of a proof?